Finite Summation of Integer Powers (Part 3)

(Discrete Mathematics Techniques III)

Abstract
We find a direct closed-form solution, i.e. one that does not require iteration, for the general case of the finite-summation-of-integer-powers problem S_p(N) = \sum_{k=1}^{N} k^p. Having established in Part 2 that the closed-form solution is a polynomial, the summation is here rewritten as the sum of the p+1 independent monomials a_j N^j (1 \leq j \leq p+1), where the a_j are unknown coefficients. Using the recurrence relation S_p(N+1) = S_p(N) + (N+1)^p, we obtain a linear combination of the monomials, which reduces to an easily solvable (p+1)-by-(p+1) triangular linear system in the unknown coefficients a_j of the closed-form polynomial solution. Maxima and Octave/Matlab codes for directly computing the closed-form solutions are included in the Appendices.

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Why Zero Raised to the Zero Power IS One

Updated! February 5, 2017

The value of zero raised to the zero power, (0^0), has been discussed since the time of Euler in the 18th century (1700s). There are three reasonable choices: 1,0, or “indeterminate”. Despite consensus amongst mathematicians that the correct answer is one, computing platforms seem to have reached a variety of conclusions: Google, R, Octave, Ruby, and Microsoft Calculator choose 1; Hexelon Max and TI-36 calculator choose 0; and Maxima and Excel throw an error (indeterminate). In this article, I’ll explain why, for discrete mathematics, the correct answer cannot be anything other than 0^0=1, for reasons that go beyond consistency with the Binomial Theorem (Knuth’s argument).
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Finite Summation of Integer Powers (Part 2)

(Discrete Mathematics Techniques II)

Abstract
We solve the general case of the finite-summation-of-integer-powers problem S_p(N) = \sum_{k=1}^{N} k^p for arbitrary p, and obtain a p-th order recurrence relation that can be used to iteratively obtain the closed form polynomial for S_p(N) for any given p. Source code is given for computing these polynomials using Maxima, an open-source (free) symbolic computation platform. (Note: This article generalizes the recurrence relation approach that is motivated and illustrated for small p in Part 1. A direct matrix method for computing the closed form solutions is given in Part 3.)

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Finite Summation of Integer Powers (Part 1)

(Discrete Mathematics Techniques I)

Abstract
We motivate an approach that uses recurrence relations to find closed form solutions to the finite-summation-of-integer-powers problem S_p(N) = \sum_{k=1}^{N} k^p for any individual p. The approach is illustrated for small p: k, k^2, k^3, k^4. Maxima, an open-source (free) software package, is used to demonstrate how a symbolic computation platform can speed up the accurate derivation of messy algebraic expressions.

A recurrence solution to the general case (arbitrary p) is developed in Part 2 along with Maxima source code. A direct (non-iterative) matrix method for solving the general case is given in Part 3 along with Maxima and Octave/Matlab source code.

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Good mathematical technique and the case for mathematical insight

Good mathematical technique can bring the solution to certain mathematical questions within reach. By a proper formulation (one that is both tractable and that generalizes readily) and the use of mechanical techniques, one can often pass from a single insight to the solution of a family of problems, and in some cases, to the solution of the general question itself. … Good mathematical technique has built within it the mathematical insight of the best of previous generations.

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